We will focus on the two-step phase transition shown in Fig. REF -(b).
According to this phase transition path, a global symmetry in the potential is broken during the phase transition.
After the phase transition, the broken symmetry can be restored in the current universe.
Such a multi-step phase transition is significant to explain the phenomena beyond the standard model; dark matter existence [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, baryon asymmetry of the universe [18]}, [19]}, [20]}, [21]}, [22]}, [23]}, [24]}, [25]}, [26]}, [27]}, [28]}, [29]}, [30]}, [31]}, [32]}, [33]}.
Furthermore, the phase transition in the second path of Fig. REF -(b) can be strong because the barrier shows up in the potential at the zero temperature by tree-level effects, like Ref. [19]}. We will show the conditions in which the two-step phase transition happens in this general model. For the EWPT in Fig. REF -(b), the first (second) step relies on the thermal loop (tree-level) effects, and thus the sizable barrier could be generated in the second step by the tree-level effects. The detectable GW spectrum is mainly produced by the second step since the tree-level effect is the dominant contribution to this phase transition path. On the other hand, the first-order EWPT in Fig. REF -(a) can be realized by finite thermal effects only. Since the sources of the barrier are different between the one-step and the two-step phase transitions, the temperature for starting the EWPT of the Fig. REF -(a) is higher than that of the Fig. REF -(b).
Therefore, we may distinguish the patterns of the EWPT in Fig. REF -(a) and -(b) by the GW observation.
